An ortho-base is a base ℬ such
that the intersection of any subcollection is either an open set or a singleton for
which the subcollection is a local base. This paper is primarily devoted to the
relationship of bases of this sort to other topological properties of bases, and of the
space itself. In §1, we compare ortho-bases to bases having similar properties,
and study the relationship with developability and paracompactness. In
§2, we define “rank” of a base for arbitrary cardinals and show how it is
related to orthocompactness, ortho-bases, and bases of countable order.
Section 3 treats two related ascending chain conditions in relation to bases of
sub-infinite rank and ortho-bases. Section 4 relates the possession of an ortho-base
to a number of “generalized metrlc” properties such as first countability,
quasi-developability, and quasi-metrizability. The remaining two sections give
examples illustrating the various properties and raise a number of unsolved
problems.
